Chapter 2 Statistical Descriptions of Device Variation
2.4 The Distribution of Initial Parameter Difference
3.1. Simulation Method Review 3.1.1 Worst Case Method 3.1.2 Monte Carlo Method
3.2. The Simulation Techniques of Device Variation 3.3. Results
Chapter 4. Effects of Simulation Techniques on TFTs Circuits Performance 4.1. Ring Oscillator
4.2. Differential Pair
Chapter 5. Conclusions and Future Works References
Fig. 1.1 The block diagram of an active matrix display
Fig. 1.2 The integration of peripheral circuits in a display achieved by poly-Si TFTs
Data In
Fig. 1.3 The display panel system
Fig. 1.4 The initial characteristics of LTPS TFTs are different from one another due to various distributions of grain boundaries
Fig. 1.5 The site variation of the threshold voltage variation for LTPS TFT fabrication line plotted in the format of lot trend
Chapter 2
Statistical Descriptions of Device Variation
2.1. Introduction to crosstie TFTs and Device Fabrication
In prior studies, it is known that LTPS TFTs suffered from severe device variation even under well-controlled process. Since the device variation is inevitable in LTPS TFTs, it is essential to classify the sources of variation. In MOSFETs (Metal-Oxide-Silicon Field Effect Transistors), the local variations can be characterized by short correlation distances and global variations characterized by long correlation distances, where the correlation distance is defined as the distance in which a process disturbance affects the device performances. If this distance is lower than the usual distance between devices, the disturbance constitutes a local variation and affects few devices (e.g. a charge trapped in the gate oxide layer). For the global variation, which is characterized by process disturbances with longer correlation distances (e.g. the gate oxide thickness across the wafer surface), affects all the devices within a defined region. Therefore, the devices placed at longer distance are more affected by global variations than devices placed close to each other.
In order to investigate the relationship between uniformity issue and device distance, a special layout of the devices adopted in this work is shown in Fig 2.1. The red, blue and yellow regions respectively represent the polysilicon film, the gate metal and the source/drain metal. The structure of the poly-Si film and the gate metal are in the order that resembles the crosstie of the railroad and therefore this layout is called the crosstie type layout of LTPS TFTs. The distance of two nearest active regions is equally-spaced 40µm. The global variation may be ignored within this small distance, and the variation of device behavior can therefore be reduced to only local variation.
For this reason, we can find out the relationship between the variation behaviors and the distance of mutual devices by adopting the crosstie layout TFTs.
The process flow of TFTs is described below. Top gate LTPS TFTs with width/length dimension of 20 µm / 5 µm were fabricated using low temperature process. Firstly, the buffer oxide and a-Si:H film with thickness of 50 nm were deposited on glass substrates with PECVD. The samples were then put in the oven for dehydrogenation. The XeCl excimer laser of wavelength 308 nm and energy density of 400 mJ/cm2 was applied. The laser scanned the a-Si:H film with the beam width of 4 mm and 98% overlap to recrystallize the a-Si:H film to poly-Si. After poly-Si active area definition, 100 nm SiO2 was deposited with PECVD as the gate insulator. Next, the metal gate was formed by sputter and then defined. The lightly doped drain (LDD) and the n+ source/drain doping were formed by PH3 implantation with dosage 2 × 1013 cm-2 and 2 × 1015 cm-2 of PH3 respectively. The LDD implantation was self-aligned and the n+ regions were defined with a separate mask. Then, the interlayer of SiNX
was deposited. Subsequently, the rapid thermal annealing was conducted to activate the dopants. Meanwhile, the poly-Si film was hydrogenated. Finally, the contact hole formation and metallization were performed to complete the fabrication work. The Fig. 2.2 shows the schematic cross-section structure of the n-type poly-Si TFT with lightly doped drain (LDD).
2.2. Statistical Descriptions of Device Variation
The variability of the observations in a data set is often another important feature of interest when data sets are summarized. We now consider several summary measures of variability, sometimes also called measures of dispersion or spread.
2.2.1. Variance and Standard deviation
The most commonly used measure of variability in statistical analysis is called the variance. It is a measure that takes into account all the observations in a data set.
The variance s2 is expressed in units that are the square of the units of measure of the variable under study. Take the positive square root of the variance, and the resulting value is called the standard deviation and is also used as a measure of variability.
2.2.2. Range
The range is the difference between the largest and smallest observations in a data set. A limitation of the range as a measure of the variability of a data set is that it depends only on the largest and smallest observations quite close to each other with the exception of one outlying observation. Despite the concentration of almost all the observations, the range would be large because of the one outlying observation.
Another limitation of the range as a measure of variability is that it is affected by the number of observations in the data set. The larger the number of observations, the larger the range tends to be. Sometimes, the range of a data set is indicated by presenting the smallest and largest observations in the data set. This form of presentation not only provides information about the variability of a data set but also provides information about the location of the data set distribution.
2.2.3. Inter-quartile Range
Because the range depends only on the smallest and largest observations in a data set, a modified range is sometimes used that reflects the variability of the middle 50 percent of the observations in the array. This modified range is called the inter-quartile range. The inter-quartile range is the difference between the third and first quartiles of the data set. The inter-quartile range may be considered to be approximately the range for a trimmed data set in which the smallest 25 percent and
the largest 25 percent of observations have been removed.
2.3. The Distribution of Initial Parameter
Firstly, we introduce the statistical expressions for the following analysis. The average value µ is defined as
The standard deviation value, σ, is usually used to investigate the distribution of the observed value. The standard deviation value is given as
( )
21
n
x X
σ ≡ n
∑
− Where x is the observe value (2-2) In order to obtain the more accurate parameter distributions of crosstie layout TFTs, large amount of device parameters are required. In this work, more than six hundred of devices were measured within 45µm on the glass substrate. The distributions of VTH and Mu of measured devices are shown respectively in Fig. 2.3 (a), (b), and Fig. 2.4 (a), (b). The average and standard deviation of VTH of n-type are 1.69 V and 0.03 V, and those of p-type are -2.41 V and 0.05 V. On the other hand, the average and standard deviation of Mu of n-type are 59.66 cm2 /Vs and 7.84 cm2/Vs, and those of p-type are 75.31 cm2 /Vs and 2.29 cm2/Vs, accordingly.These figures reveal that the distributions of VTH and Mu are asymmetric and non-Gaussian, and there is no fit equation to represent the variation. In particular, the distributions of Mu have higher degree of skew than VTH. Compared p-type with n-type devices, the distributions of VTH and Mu are both less asymmetric in the p-type than in the n-type devices. Especially, the distribution of VTH of the p-type devices approximates to Gaussian distribution. However, it is so difficult to model the variable behavior that the circuit simulation can not be put into practice.
Let me take a look in Fig. 2.5, it indicates that the distributions of initial parameters vary with the different sites on glass and lot. If we want to find the variation behaviors with respect to the distance, it can not just classify them via these distributions. Another grouping method mentioned in the next section will get the more identical distributions, which will be more useful to evaluate the variations in LTPS TFTs.
2.4. The Distribution of Initial Parameter Difference
Return to analyze initial VTH and Mu of the measured devices with position, those are shown respectively in Fig. 2.6 (a) and (b).
Fig. 2.6 (a) and (b) reveal VTH and Mu of the measured devices vary irregularly with position, and the variable behaviors are like signal and noise. The noise such as the micro variation is the distributions of the difference of device parameters. The concept of the description of device parameter variation will be proposed as shown as Fig. 2.7.
The signal such as the variation in range is determined by gate insulator thickness, ion implantation dosage, channel length, LDD length and so on. The noise such as the micro variation is determined by defect sites, defect density, activation efficiency and so on. If it can find that fit equations used to describe the distributions of the difference of device parameters, the noise in circuit simulation will randomly generate from them. Therefore the distributions of the difference of VTH and Mu for device pairs of the measured devices are described below.
In order to identify the effects of the global and local variation, the parameters differences of two devices under certain distance are divided with several groups according to the distance between two devices. In prior studies, the averages of parameters differences stand for global variation of LTPS TFTs, while the standard
deviation of parameter differences shows the local variation in the devices. In this thesis, we characterize the global variation and local variation as the variation in the range and micro variation for the analysis of LTPS TFTs, respectively. Fig. 2.8 and Fig. 2.9 show the average and the standard deviation of the differences of VTH and Mu. As the mutual device distance increases, the deviations of device differences are not changing with the device distance.
It can be explained that the micro variation will merely vary with distance as we expect. As for the variation in a range, these figures show the diverse results. In the difference of VTH, the average is increasing with device distance. However, the average of the difference of Mu seems no significant trend when the distance of mutual devices is increasing. Although the averages of the differences of these parameters show different behaviors, they still appear in linear form. On the other hand, the effects of variation in a range are still minor than those of the micro variation under short device distance.
Since these conditions differ from device to device, the micro variation will lead to the random distribution of device parameters. For the circuit simulation, Monte Carlo method is generally adopted. However, the worst case simulation will be more suitable when the variation in a range of device is increasing.
In Fig. 2.10 and Fig. 2.11, the distributions of the difference of (a) VTH and (b) Mu can be used to describe the micro variation of the measured devices. According to these figures, the shape of these distributions is symmetrical and looks like Gaussian distribution. The range of these distributions is wider than triple standard deviation of Gaussian distribution so these distributions are not Gaussian distribution.
Furthermore, there are some uncommon equations similar with Gaussian distribution, and they are more suitable to describe the variation behavior. In next work, the square of the correlation coefficient (R square) presenting the fitness from the chosen
equation will be used. R square is defined as
R square which indicates the similarity between the proposed model and the real data, and its value ranges between 0 and 1 [9]. It represents the proposed model is more similar with real data when R square value much approaches to 1. Generally speaking, the values of R square above 0.7 represnent the good fitness for the chosen funcion. Two models are proposed to fit these two distributions, and the square of the correlation coefficient (R square) can reach near 0.9. Consequently, the difference of VTH follows the distribution of Gaussian-Lorentzian cross product, which is
( )
⎟⎟On the other hand, the difference of Mu follows Lorentzian distribution, which is
2
Where the parameters a, b, c, and d are fitting parameters and vary slightly with distance. These two distributions are more concentrated than the commonly known Gaussian distribution, and the peaks of these are sharper.
We polt the fitting results with different device distance in Fig. 2.12 (a) ~ (d) and Fig. 2.13 (a) ~ (d) for the distributions of the differences of Vth and Mu, respectively.
The values of R squre of the above fittng curves both reach near 0.9. It clearly shows the good fitness of our proposed mathemtical model. Most of the fitting parameters slightly changing with distance supports the effects of the variation in the range are minor than those of micro variation we mentioned before. However, we still have to notice that micro variation increasing rapidly with distance and saturate about the device distance of 2000 µm.
This finding will affect the result of the circuit simulation. For above reasons, how to apply the proposed model in circuit simulation is very critical and important.
There are several methods of parameter generation in next section.
Fig. 2.1 The layout of the crosstie TFTs
Fig. 2.2 The schematic cross-section structure of the n-type poly-Si TFT with lightly doped drain
Fig. 2.3 (a) The distributions of threshold voltage of n-type devices for crosstie TFTs
Fig. 2.3 (b) The distributions of threshold voltage of p-type devices for crosstie TFTs
Fig. 2.4 (a) The distributions of mobility of n-type for crosstie TFTs
Fig. 2.4 (b) The distributions of mobility of p-type for crosstie TFTs
Fig. 2.5 The distributions of initial parameters vary with the different sites on glass and lot
Fig. 2.6 (a) The variations of VTH of the measured devices with position
Fig. 2.6 (b) The variations of mobility of the measured devices with position
Fig. 2.7 The concept of the proposed description of device parameter Vth
position
= +
the variation in the range micro variation
Fig. 2.8 The average and the standard deviation of the differences of threshold voltage
Fig. 2.9 The average and the standard deviation of the differences of mobility
Fig. 2.10 (a) The distribution of VTH difference of n-type devices and its fitting curve under the device distance of 40 µm
Fig. 2.10 (b) The distribution of VTH difference of p-type devices and its fitting curve under the device distance of 40 µm
Fig. 2.11 (a) The distribution of Mu difference of n-type devices and its fitting curve under the device distance of 40 µm
Fig. 2.11 (b) The distribution of Mu difference of p-type devices and its fitting curve under the device distance of 40 µm
Fig. 2.12 (a) The distribution of VTH difference of n-type devices and its fitting curve under the device distance of 200 µm
Fig. 2.12 (b) The distribution of VTH difference of n-type devices and its fitting curve under the device distance of 2000 µm
Fig. 2.12 (c) The distribution of VTH difference of p-type devices and its fitting curve under the device distance of 200 µm
Fig. 2.12 (d) The distribution of VTH difference of n-type devices and its fitting curve under the device distance of 2000 µm
Fig. 2.13 (a) The distribution of Mu difference of n-type devices and its fitting curve under the device distance of 200 µm
Fig. 2.13 (b) The distribution of Mu difference of n-type devices and its fitting curve under the device distance of 2000 µm
Fig. 2.13 (c) The distribution of Mu difference of p-type devices and its fitting curve under the device distance of 200 µm
Fig. 2.13 (d) The distribution of Mu difference of p-type devices and its fitting curve under the device distance of 2000 µm
Chapter 3
Simulation Techniques of Device Variation
3.1. Simulation Methods Review
There are two major methods of simulation to analyze circuit performance, which are the worst-case and Monte Carlo analysis as described below [11].
3.1.1. Worst-Case Method
Worst-Case analysis is the most commonly used technique in industry for considering manufacturing process tolerances in the design of integrated circuits.
These approaches are relatively inexpensive compared to the yield maximization approaches in terms of computational cost and designer effort, and they also provide high parametric yields. At any design point, uncontrollable fluctuations in the circuit parameters cause circuit performance to device from their nominal design values. The goal of worst case analysis is to determine the worst values that the performance may have under these statistical fluctuation. In addition to finding the worst-case values of the circuit performance, this analysis also finds the corresponding worst-case values of noise parameters.A noise parameter is treated as a random variable. Any random variable is characterized by probability density function (and by a mean and a standard deviation which depends on the density function), as shown in Fig. 3.1. The worst-case noise parameter vector is used in circuit simulation to verify whether circuit performances are acceptable under these conditions. Similar to worst-case analysis, one can also perform best-case analysis. In fact, industrial designs are often simulated under best, worst, and nominal noise parameter conditions, which provide designers with quick estimates of range of variation of circuit performances.
3.1.2. Monte Carlo Method
Yield, expressed as a multi-dimensional integral, can be evaluated numerically using either the quadrature-based, or Monte Carlo based methods. The quadrature-based methods have computational costs that explode exponentially with the dimensionality of the statistical space. Monte Carlo methods, on the other hand, are less sensitive to the dimensionality. The Monte Carlo method is a computer simulation of real distributions of random noise parameters, and it is the simplest, most reliable and accurate of all methods used in practice, but for high accuracy it requires a large number of sample points. Typically, hundreds of trials are required to obtain reasonable accurate yield estimation. For nonlinear and/or time domain circuit analysis, this is computational expensive. Hence, a fundamental problem to solve is to increase the efficiency of the Monte Carlo method and its accuracy, measured by the variance of the yield estimation.
3.2. The Simulation Techniques of Device Variation
There are two major methods of simulation to analyze circuit performance, which are the worst-case and Monte Carlo analysis as described below.
The notations of the parameters generations are shown in Fig. 3.2. There are four kind descriptions of the parameters generations.
First, it is picking up the parameters randomly from the measured database, and calls “Look up table.” Look up table (LUT) is most direct method, but it is complicated and costs much time. The concept of look up table parameters generation is shown in Fig. 3.3. But a ring oscillator is a special case, and it needs to choose one series of initial parameter in order as 1st class and randomly choose another series of initial parameter in order as 2nd class.
Second, the Monte Carlo simulation of Gaussian distribution is a conventional method [10]. It needs to calculate mean and standard deviation of the measured data, and picks up the parameters randomly from Gaussian distribution based on mean and with triple standard deviation range. In general, Gaussian distribution is commonly used for circuit simulation.
However, the behavior of the variation is different from Gaussian distribution, and the conventional method might not suitable enough to simulate for the circuits performance with device variation. Third, modified Gaussian distribution which is
However, the behavior of the variation is different from Gaussian distribution, and the conventional method might not suitable enough to simulate for the circuits performance with device variation. Third, modified Gaussian distribution which is